I want to find all ways that the group $A_5$ acting on a set of 3 elements and what can be the orbit size?
My attempt: I thinking it is same as finding all the homomorphism into $S_3$. $S_3$ has six elements and it is non-abelian.
The group $A_5$ is simple group so there is no nontrivial normal subgroup. Kernel of homomorphism is normal so it's either the identity or the whole group $A_5$.
So if there is a homomorphism, then it's either injective or trivial.
Since $A_5$ has $60$ elements then any homomorphism is trivial. So the action is trivial.
The orbit size in this case is one.
Is my answer correct?